Holonomic splitting

Question

I am reading the book "Introduction to the h-Principle" by Eliashberg and Mishachev. At the moment I try to understand the Section 1.7 Holonomic splitting on page 12 but without success. I do not understand the Holonomic splitting proposition. What does it exactly say? Can one put this in mathematical language? I think it means that over some sufficiently small ball $U\subset V$ one can write the bundle $X^{(r)}\rightarrow V$ as the trivial fibration $U\times \mathbb{R}^{qN_{r}} \rightarrow U$. What I do not understand is why one needs a holonomic section for that? Hope there is somebody who can explain those things to me.